A priori Hölder estimates for equations degenerating on nodal sets
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918104233148416 |
|---|---|
| author | Terracini, Susanna Tortone, Giorgio Vita, Stefano |
| author_facet | Terracini, Susanna Tortone, Giorgio Vita, Stefano |
| contents | We prove a priori Hölder bounds for continuous solutions to degenerate equations with variable coefficients of type $$ \mathrm{div}\left(u^2 A\nabla w\right)=0\quad\mathrm{in \ }Ω\subset\mathbb R^n,\qquad \mbox{with}\qquad \mathrm{div}\left(A\nabla u\right)=0, $$ where $A$ is a Lipschitz continuous, uniformly elliptic matrix (possibly $u$ has non-trivial singular nodal set). Such estimates are uniform with respect to $u$ in a class of normalized solutions that have a bounded Almgren frequency. As a consequence, a boundary Harnack principle holds for the quotient of two solutions vanishing on a common set.
This analysis relies on a detailed study of the associated weighted Sobolev spaces, including integrability of the weight, capacitary properties of the nodal set, and uniform Sobolev inequalities yielding local boundedness of solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_18991 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A priori Hölder estimates for equations degenerating on nodal sets Terracini, Susanna Tortone, Giorgio Vita, Stefano Analysis of PDEs We prove a priori Hölder bounds for continuous solutions to degenerate equations with variable coefficients of type $$ \mathrm{div}\left(u^2 A\nabla w\right)=0\quad\mathrm{in \ }Ω\subset\mathbb R^n,\qquad \mbox{with}\qquad \mathrm{div}\left(A\nabla u\right)=0, $$ where $A$ is a Lipschitz continuous, uniformly elliptic matrix (possibly $u$ has non-trivial singular nodal set). Such estimates are uniform with respect to $u$ in a class of normalized solutions that have a bounded Almgren frequency. As a consequence, a boundary Harnack principle holds for the quotient of two solutions vanishing on a common set. This analysis relies on a detailed study of the associated weighted Sobolev spaces, including integrability of the weight, capacitary properties of the nodal set, and uniform Sobolev inequalities yielding local boundedness of solutions. |
| title | A priori Hölder estimates for equations degenerating on nodal sets |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2507.18991 |